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Is there a way to somehow establish a connection between two users on a website so that they can IM each other in real time?

Facebook has this ;)

Right, but I don't own Facebook But that's the idea.
Is there some sort of widget I can integrate into my website?

Take a look at hotscripts.com and see what they have floating about there.

I've tested this once, and I liked it.....
http://hot-things.net/blab-im-ajax-php-messenger

I've got an IRC going at the moment, private chat is allowed.

oakhill.x10hosting.com/IRC.html

Ajax IM isn't bad http://ajaxim.com/

http://mibbit.com
AjAX IRC client with widget support.

or you can use google wave >_<